Abstract

The measurement of the refractive index n and the physical thickness t of a homogeneous thin film by ellipsometry is a well-established technique. Ellipsometry also opens up the possibility of characterizing inhomogeneous thin films. This paper deals with the theoretical analysis of the sensitivity of ellipsometry to a linear refractive index gradient within a single-layer thin film. A major result is that, to a first approximation, the ellipsometric data for a film at a wavelength that is an odd multiple of four times the optical thickness (i.e., at the odd quarter-wave points) are unaffected by the inhomogeneity of the film but depend only on the average refractive index of the film. Ellipsometric data for wavelengths that are multiples of twice the optical thickness (the half-wave points) are strongly affected by the inhomogeneity of the films. Results are presented relative to the sensitivity of ellipsometric measurements for the determination of the average index and the degree of inhomogeneity of a film.

© 1990 Optical Society of America

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  1. D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Dobrowolski, U. J. Gibson, T. Tuttle-Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A. Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H. Strome, R. Swenson, P. A. Temple, T. F. Thonn, “Multiple determination of the optical constants of thin-film coating materials,” Appl. Opt. 23, 3571–3596 (1984).
    [CrossRef] [PubMed]
  2. D. E. Aspnes, H. G. Craighead, “Multiple determination of the optical constants of thin-film coating materials: an Rh sequel,” Appl. Opt. 25, 1299–1310 (1986).
    [CrossRef] [PubMed]
  3. M. Harris, H. A. Macleod, S. Ogura, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
    [CrossRef]
  4. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
    [CrossRef] [PubMed]
  5. B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the optical properties of some high-index oxide films prepared by ion-beam sputter deposition with those of electronbeam evaporated films,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 410–417 (1986).
  6. J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
    [CrossRef] [PubMed]
  7. F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
    [CrossRef]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1977), Chap. 4.
  9. P. J. McMarr, K. Vedam, “Spectroscopic ellipsometer: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
    [CrossRef]
  10. H. Arwin, D. E. Aspnes, “Unambiguous determination of thickness and dielectric function of thin films by spectroscopic ellipsometry,” Thin Solid Films 113, 101–113 (1984).
    [CrossRef]
  11. J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3773–3779 (1985).
    [CrossRef] [PubMed]
  12. C. Dhanavantri, R. N. Karekar, V. J. Rao, “Study of graded aluminum oxide films prepared by metal-organic chemical vapour deposition,” Thin Solid Films 127, 85–91 (1985).
    [CrossRef]
  13. R. W. Collins, “In situ investigation of the nucleation of microcrystaline Si,” Appl. Phys. Lett. 48, 843–845 (1986).
    [CrossRef]
  14. R. W. Collins, J. M. Cavese, “Effect of deposition conditions on the nucleation and growth of glow discharge a-Si:H,” J. Appl. Phys. 61, 1869–1882 (1987).
    [CrossRef]
  15. F. Hottier, J. B. Theetan, “Surface analysis during vapor-phase growth,” J. Cryst. Growth 48, 644–654 (1980).
    [CrossRef]
  16. R. W. Collins, “Surface, interface, and bulk properties of amorphous carbon films characterized by in situ ellipsometry,” Appl. Phys. Lett. 52, 2025–2027 (1988).
    [CrossRef]
  17. R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, p. 247.
    [CrossRef]
  18. H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichlige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).
  19. R. Jacobsson, “Inhomogeneities of co-evaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–98.
  20. H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 21et seq.
  21. C. K. Carniglia, “Method for measuring the optical properties of slightly absorbing inhomogeneous dielectric thin films,” J. Opt. Soc. Am. A 3 (13), P 40 (1986).

1989

1988

R. W. Collins, “Surface, interface, and bulk properties of amorphous carbon films characterized by in situ ellipsometry,” Appl. Phys. Lett. 52, 2025–2027 (1988).
[CrossRef]

1987

R. W. Collins, J. M. Cavese, “Effect of deposition conditions on the nucleation and growth of glow discharge a-Si:H,” J. Appl. Phys. 61, 1869–1882 (1987).
[CrossRef]

1986

R. W. Collins, “In situ investigation of the nucleation of microcrystaline Si,” Appl. Phys. Lett. 48, 843–845 (1986).
[CrossRef]

P. J. McMarr, K. Vedam, “Spectroscopic ellipsometer: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the optical properties of some high-index oxide films prepared by ion-beam sputter deposition with those of electronbeam evaporated films,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 410–417 (1986).

C. K. Carniglia, “Method for measuring the optical properties of slightly absorbing inhomogeneous dielectric thin films,” J. Opt. Soc. Am. A 3 (13), P 40 (1986).

D. E. Aspnes, H. G. Craighead, “Multiple determination of the optical constants of thin-film coating materials: an Rh sequel,” Appl. Opt. 25, 1299–1310 (1986).
[CrossRef] [PubMed]

1985

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3773–3779 (1985).
[CrossRef] [PubMed]

C. Dhanavantri, R. N. Karekar, V. J. Rao, “Study of graded aluminum oxide films prepared by metal-organic chemical vapour deposition,” Thin Solid Films 127, 85–91 (1985).
[CrossRef]

1984

1982

1980

F. Hottier, J. B. Theetan, “Surface analysis during vapor-phase growth,” J. Cryst. Growth 48, 644–654 (1980).
[CrossRef]

1979

M. Harris, H. A. Macleod, S. Ogura, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

1963

F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
[CrossRef]

1941

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichlige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).

Albrand, G.

Allen, T. H.

Arndt, D. P.

Arwin, H.

H. Arwin, D. E. Aspnes, “Unambiguous determination of thickness and dielectric function of thin films by spectroscopic ellipsometry,” Thin Solid Films 113, 101–113 (1984).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, H. G. Craighead, “Multiple determination of the optical constants of thin-film coating materials: an Rh sequel,” Appl. Opt. 25, 1299–1310 (1986).
[CrossRef] [PubMed]

H. Arwin, D. E. Aspnes, “Unambiguous determination of thickness and dielectric function of thin films by spectroscopic ellipsometry,” Thin Solid Films 113, 101–113 (1984).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1977), Chap. 4.

Bennett, J. M.

Blanco, J. R.

Borgogno, J. P.

Carniglia, C. K.

Case, W. E.

Cavese, J. M.

R. W. Collins, J. M. Cavese, “Effect of deposition conditions on the nucleation and growth of glow discharge a-Si:H,” J. Appl. Phys. 61, 1869–1882 (1987).
[CrossRef]

Collins, R. W.

R. W. Collins, “Surface, interface, and bulk properties of amorphous carbon films characterized by in situ ellipsometry,” Appl. Phys. Lett. 52, 2025–2027 (1988).
[CrossRef]

R. W. Collins, J. M. Cavese, “Effect of deposition conditions on the nucleation and growth of glow discharge a-Si:H,” J. Appl. Phys. 61, 1869–1882 (1987).
[CrossRef]

R. W. Collins, “In situ investigation of the nucleation of microcrystaline Si,” Appl. Phys. Lett. 48, 843–845 (1986).
[CrossRef]

Craighead, H. G.

Dhanavantri, C.

C. Dhanavantri, R. N. Karekar, V. J. Rao, “Study of graded aluminum oxide films prepared by metal-organic chemical vapour deposition,” Thin Solid Films 127, 85–91 (1985).
[CrossRef]

Dobrowolski, J. A.

Gibson, U. J.

Guenther, K. H.

Harris, M.

M. Harris, H. A. Macleod, S. Ogura, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Ho, F. C.

Hodgkin, V. A.

Hottier, F.

F. Hottier, J. B. Theetan, “Surface analysis during vapor-phase growth,” J. Cryst. Growth 48, 644–654 (1980).
[CrossRef]

Jacobsson, R.

R. Jacobsson, “Inhomogeneities of co-evaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–98.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, p. 247.
[CrossRef]

Karekar, R. N.

C. Dhanavantri, R. N. Karekar, V. J. Rao, “Study of graded aluminum oxide films prepared by metal-organic chemical vapour deposition,” Thin Solid Films 127, 85–91 (1985).
[CrossRef]

Klapp, W. P.

Lazarides, B.

Macleod, H. A.

McCrackin, F. L.

F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
[CrossRef]

McMarr, P. J.

P. J. McMarr, K. Vedam, “Spectroscopic ellipsometer: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3773–3779 (1985).
[CrossRef] [PubMed]

Ogura, S.

M. Harris, H. A. Macleod, S. Ogura, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

Passaglia, E.

F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
[CrossRef]

Pelletier, E.

Pond, B.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the optical properties of some high-index oxide films prepared by ion-beam sputter deposition with those of electronbeam evaporated films,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 410–417 (1986).

Purvis, M. K.

Quinn, D. M.

Raj, T.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the optical properties of some high-index oxide films prepared by ion-beam sputter deposition with those of electronbeam evaporated films,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 410–417 (1986).

Rao, V. J.

C. Dhanavantri, R. N. Karekar, V. J. Rao, “Study of graded aluminum oxide films prepared by metal-organic chemical vapour deposition,” Thin Solid Films 127, 85–91 (1985).
[CrossRef]

Saxer, A.

Schmell, R. A.

J. M. Bennett, E. Pelletier, G. Albrand, J. P. Borgogno, B. Lazarides, C. K. Carniglia, R. A. Schmell, T. H. Allen, T. Tuttle-Hart, K. H. Guenther, A. Saxer, “Comparison of the properties of titanium dioxide films prepared by various techniques,” Appl. Opt. 28, 3303–3317 (1989).
[CrossRef] [PubMed]

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the optical properties of some high-index oxide films prepared by ion-beam sputter deposition with those of electronbeam evaporated films,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 410–417 (1986).

Schröder, H.

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichlige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).

Steinberg, H. L.

F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
[CrossRef]

Strome, D. H.

Stronberg, R. R.

F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
[CrossRef]

Swenson, R.

Temple, P. A.

Theetan, J. B.

F. Hottier, J. B. Theetan, “Surface analysis during vapor-phase growth,” J. Cryst. Growth 48, 644–654 (1980).
[CrossRef]

Thonn, T. F.

Tuttle-Hart, T.

Vedam, K.

P. J. McMarr, K. Vedam, “Spectroscopic ellipsometer: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

J. R. Blanco, P. J. McMarr, K. Vedam, “Roughness measurements by spectroscopic ellipsometry,” Appl. Opt. 24, 3773–3779 (1985).
[CrossRef] [PubMed]

Ann. Phys. (Leipzig)

H. Schröder, “Bemerkung zur Theorie des Lichtdurchgangs durch inhomogene durchsichlige Schichten,” Ann. Phys. (Leipzig) 39, 55–58 (1941).

Appl. Opt.

Appl. Phys. Lett.

R. W. Collins, “In situ investigation of the nucleation of microcrystaline Si,” Appl. Phys. Lett. 48, 843–845 (1986).
[CrossRef]

R. W. Collins, “Surface, interface, and bulk properties of amorphous carbon films characterized by in situ ellipsometry,” Appl. Phys. Lett. 52, 2025–2027 (1988).
[CrossRef]

J. Appl. Phys.

R. W. Collins, J. M. Cavese, “Effect of deposition conditions on the nucleation and growth of glow discharge a-Si:H,” J. Appl. Phys. 61, 1869–1882 (1987).
[CrossRef]

P. J. McMarr, K. Vedam, “Spectroscopic ellipsometer: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

J. Cryst. Growth

F. Hottier, J. B. Theetan, “Surface analysis during vapor-phase growth,” J. Cryst. Growth 48, 644–654 (1980).
[CrossRef]

J. Opt. Soc. Am. A

J. Res. Natl. Bur. Stand. (U.S.)

F. L. McCrackin, E. Passaglia, R. R. Stronberg, H. L. Steinberg, “Measurement of the thickness and refractive index of very thin films and the optical properties of surfaces by ellipsometry,” J. Res. Natl. Bur. Stand. (U.S.) 67A, 363–377 (1963).
[CrossRef]

Natl. Inst. Stand. Technol. Spec. Publ.

B. Pond, R. A. Schmell, C. K. Carniglia, T. Raj, “Comparison of the optical properties of some high-index oxide films prepared by ion-beam sputter deposition with those of electronbeam evaporated films,” Natl. Inst. Stand. Technol. Spec. Publ. 752, 410–417 (1986).

Thin Solid Films

M. Harris, H. A. Macleod, S. Ogura, “The relationship between optical inhomogeneity and film structure,” Thin Solid Films 57, 173–178 (1979).
[CrossRef]

H. Arwin, D. E. Aspnes, “Unambiguous determination of thickness and dielectric function of thin films by spectroscopic ellipsometry,” Thin Solid Films 113, 101–113 (1984).
[CrossRef]

C. Dhanavantri, R. N. Karekar, V. J. Rao, “Study of graded aluminum oxide films prepared by metal-organic chemical vapour deposition,” Thin Solid Films 127, 85–91 (1985).
[CrossRef]

Other

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Oxford, 1977), Chap. 4.

R. Jacobsson, “Light reflection from films of continuously varying refractive index,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1966), Vol. 5, p. 247.
[CrossRef]

R. Jacobsson, “Inhomogeneities of co-evaporated homogeneous films for optical applications,” in Physics of Thin Films, G. Hass, M. H. Francombe, R. W. Hoffman, eds. (Academic, New York, 1975), Vol. 8, pp. 51–98.

H. A. Macleod, Thin Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 21et seq.

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Figures (10)

Fig. 1
Fig. 1

Typical refractive-index profile of an inhomogeneous single-layer film located between z = 0 and z = t.

Fig. 2
Fig. 2

Plot of Ψ versus Δ for homogeneous single-layer thin films on fused silica (ns = 1.45) for an incidence angle of 60°. The curves are for film indices ranging from 1.3 to 2.5 as indicated on the figure.

Fig. 3
Fig. 3

Plot of Ψ versus Δ for a homogeneous single layer with n = 2 at various incidence angles. The point at which the film has an optical thickness that is an integer multiple of a HW is indicated by an arrowhead pointing in the direction of increasing t/λ. a, The incidence angles are from 40° to 54°, and the curves are cyclical around a value of Δ = 180°. b, The incidence angles are from 56° to 68°, and the curves are periodic in Ψ. c, The incidence angles are from 70° to 80°, and the curves are cyclical around a value of Δ = 0°.

Fig. 4
Fig. 4

Plot of Ψ versus incidence angle for an uncoated silica substrate. This curve also corresponds to the value of Ψ at the HW points for homogeneous films as indicated by the arrowheads in Figs. 3a–3c.

Fig. 5
Fig. 5

Plot of Ψ at the QW point versus index of refraction of a homogeneous thin film for various incidence angles. Values to the left of the point at which Ψ = 0° correspond to values of Δ = 0°; values to the right correspond to Δ = 180°.

Fig. 6
Fig. 6

Incidence angle at which Ψ is 0 for the QW point versus refractive index of a single-layer homogeneous film.

Fig. 7
Fig. 7

Sensitivity of Ψ at the QW point to the refractive index of a homogeneous thin film as a function of incidence angle. The tic marks correspond to angles where Ψ(QW) = 0.

Fig. 8
Fig. 8

Plot of Ψ versus Δ for an inhomogeneous film with an average refractive index nav = 2 and various values of the degree of inhomogeneity Δn/nav from −0.2 to 0.2 as indicated. A value of Δn = 0 corresponds to a homogeneous film. The curves correspond to films with increasing t/λ starting at zero at the value Δ = 0°.

Fig. 9
Fig. 9

Sensitivity of Ψ at the HW point to the degree of inhomogeneity as a function of incidence angle. A value of Δn/nav = 0.01 was used to calculate the curves.

Fig. 10
Fig. 10

Incidence angle at which Ψ is 0 for the HW point versus degree of inhomogeneity Δn/nav for an inhomogeneous film.

Equations (34)

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n ( z ) sin [ θ ( z ) ] = n 0 sin θ 0 .
n i sin θ i = n 0 sin θ 0
r j r j 01 + r j 23 e i ϕ 1 + r j 01 r j 23 e i ϕ ,
r j 01 = ± ( n j 0 n j 1 ) / ( n j 0 + n j 1 )
r j 23 = ± ( n j 2 n j 3 ) / ( n j 2 + n j 3 ) .
n s i = n i cos θ i ,
n p i = n i / cos θ i .
ϕ = 4 π n av t cos ( θ av ) / λ ,
n av cos θ av = ( 1 / t ) 0 t n ( z ) cos [ θ ( z ) ] d z .
n ( z ) cos [ θ ( z ) ] = [ n 2 ( z ) n 0 2 sin 2 θ 0 ] 1 / 2
n i cos θ i = ( n i 2 n 0 2 sin 2 θ 0 ) 1 / 2
r j = | r j | e i δ j ,
Ψ = arctan ( | r p | / | r s | )
Δ = δ p δ s .
ϕ = m π for odd m ,
n a υ t cos θ a υ = m λ / 4 for odd m .
r j ( QW ) ( n j 1 n j 2 n j 0 n j 3 ) / ( n j 1 n j 2 + n j 0 n j 3 ) ,
ϕ = m π for even m ,
n av t cos θ av = ( m / 2 ) λ / 2 for even m .
r j ( HW ) ( n j 0 n j 2 n j 1 n j 3 ) / ( n j 0 n j 2 + n j 1 n j 3 ) .
n j 1 = n j 2 n j av
r j ( QW ) = ( n j av 2 n j 0 n j 3 ) / ( n j av 2 + n j 0 n j 3 )
r j ( HW ) = ( n j 0 n j 3 ) / ( n j 0 + n j 3 ) .
Δ n = n 1 n 2 .
n av ( n 1 + n 2 ) / 2 .
n av t cos θ av 3 λ / 4 ,
n i n av ± Δ n / 2 ,
n i cos θ i n av cos θ av [ 1 ± Δ n / ( 2 n av cos 2 θ av ) ]
n i / cos θ i ( n av / cos θ av ) { 1 ± ( Δ n / n av ) [ 1 1 / ( 2 cos 2 θ av ) ] } .
n 1 cos θ 1 n 2 cos θ 2 ( n av cos θ av ) 2
( n 1 / cos θ 1 ) ( n 2 / cos θ 2 ) ( n av / cos θ av ) 2 .
( n 2 cos θ 2 ) / n 1 cos θ 1 ) 1 Δ n / ( n av cos 2 θ av )
( n 2 / cos θ 2 ) / ( n 1 / cos θ 1 ) 1 ( Δ n / n av ) [ 2 1 / ( cos 2 θ av ) ] .
( Δ n / n av ) [ 2 1 / ( cos 2 θ av ) ] = 1 n 3 cos θ 0 / ( n 0 cos θ 3 ) .

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